The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.